From 6820542245d3943e20cb560473db81bb93599fad Mon Sep 17 00:00:00 2001 From: Claudio Sacerdoti Coen Date: Tue, 30 Jan 2007 15:18:29 +0000 Subject: [PATCH] Even more (painful) work on setoids. --- .../matita/library/technicalities/setoids.ma | 211 +++++++++++++----- 1 file changed, 158 insertions(+), 53 deletions(-) diff --git a/helm/software/matita/library/technicalities/setoids.ma b/helm/software/matita/library/technicalities/setoids.ma index f9f3d9daf..3029bc0f4 100644 --- a/helm/software/matita/library/technicalities/setoids.ma +++ b/helm/software/matita/library/technicalities/setoids.ma @@ -437,11 +437,11 @@ qed. definition get_rewrite_direction: rewrite_direction → Argument_Class → rewrite_direction. intros (dir R); cases (variance_of_argument_class R); - [ cases a; + [ exact dir + | cases a; [ exact dir (* covariant *) | exact (opposite_direction dir) (* contravariant *) ] - | exact dir (* symmetric relation *) ] qed. @@ -490,65 +490,170 @@ definition relation_of_product_of_arguments: ] qed. -(* definition apply_morphism: - ∀In Out (m: function_type_of_morphism_signature In Out) - (args: product_of_arguments In). carrier_of_relation_class Out. - intros. - induction In. - exact (m args). - simpl in m. args. - destruct args. - exact (IHIn (m c) p). + ∀In,Out.∀m: function_type_of_morphism_signature In Out. + ∀args: product_of_arguments In. carrier_of_relation_class ? Out. + intro; + elim In; + [ exact (f p) + | change in p with (Prod (carrier_of_relation_class variance t) (product_of_arguments n)); + elim p; + change in f1 with (carrier_of_relation_class variance t → function_type_of_morphism_signature n Out); + exact (f ? (f1 t1) t2) + ] qed. -Theorem apply_morphism_compatibility_Right2Left: - ∀In Out (m1 m2: function_type_of_morphism_signature In Out) - (args1 args2: product_of_arguments In). +theorem apply_morphism_compatibility_Right2Left: + ∀In,Out.∀m1,m2: function_type_of_morphism_signature In Out. + ∀args1,args2: product_of_arguments In. make_compatibility_goal_aux ? ? m1 m2 → relation_of_product_of_arguments Right2Left ? args1 args2 → directed_relation_of_relation_class Right2Left ? (apply_morphism ? ? m2 args1) (apply_morphism ? ? m1 args2). - induction In; intros. - simpl in m1. m2. args1. args2. H0 |- *. - destruct a; simpl in H; hnf in H0. - apply H; exact H0. - destruct v; simpl in H0; apply H; exact H0. - apply H; exact H0. - destruct v; simpl in H0; apply H; exact H0. - rewrite H0; apply H; exact H0. - - simpl in m1. m2. args1. args2. H0 |- *. - destruct args1; destruct args2; simpl. - destruct H0. - simpl in H. - destruct a; simpl in H. - apply IHIn. - apply H; exact H0. - exact H1. - destruct v. - apply IHIn. - apply H; exact H0. - exact H1. - apply IHIn. - apply H; exact H0. - exact H1. - apply IHIn. - apply H; exact H0. - exact H1. - destruct v. - apply IHIn. - apply H; exact H0. - exact H1. - apply IHIn. - apply H; exact H0. - exact H1. - rewrite H0; apply IHIn. - apply H. - exact H1. -Qed. + intro; + elim In; + [ simplify in m1 m2 args1 args2 ⊢ %; + change in H1 with + (directed_relation_of_argument_class + (get_rewrite_direction Right2Left t) t args1 args2); + generalize in match H1; clear H1; + generalize in match H; clear H; + generalize in match args2; clear args2; + generalize in match args1; clear args1; + generalize in match m2; clear m2; + generalize in match m1; clear m1; + elim t 0; + [ intros (T1 r Hs Hr m1 m2 args1 args2 H H1); + simplify in H; + simplify in H1; + simplify; + apply H; + exact H1 + | intros 8 (v T1 r Hr m1 m2 args1 args2); + cases v; + intros (H H1); + simplify in H1; + apply H; + exact H1 + | intros; + apply H1; + exact H2 + | intros 7 (v); + cases v; + intros (H H1); + simplify in H1; + apply H; + exact H1 + | intros; + simplify in H1; + rewrite > H1; + apply H; + exact H1 + ] + | change in m1 with + (carrier_of_relation_class variance t → + function_type_of_morphism_signature n Out); + change in m2 with + (carrier_of_relation_class variance t → + function_type_of_morphism_signature n Out); + change in args1 with + ((carrier_of_relation_class ? t) × (product_of_arguments n)); + change in args2 with + ((carrier_of_relation_class ? t) × (product_of_arguments n)); + generalize in match H2; clear H2; + elim args2 0; clear args2; + elim args1; clear args1; + elim H2; clear H2; + change in H4 with + (relation_of_product_of_arguments Right2Left n t2 t4); + change with + (relation_of_relation_class unit Out (apply_morphism n Out (m1 t3) t4) + (apply_morphism n Out (m2 t1) t2)); + generalize in match H3; clear H3; + generalize in match t3; clear t3; + generalize in match t1; clear t1; + generalize in match H1; clear H1; + generalize in match m2; clear m2; + generalize in match m1; clear m1; + elim t 0; + [ intros (T1 r Hs Hr m1 m2 H1 t1 t3 H3); + simplify in H3; + change in H1 with + (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2)); + simplify in H; + apply H; + [ apply H1; + assumption + | assumption + ] + | intro v; + elim v 0; + [ intros (T1 r Hr m1 m2 H1 t1 t3 H3); + simplify in H3; + change in H1 with + (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2)); + simplify in H; + apply H; + [ apply H1; + assumption + | assumption + ] + | intros (T1 r Hr m1 m2 H1 t1 t3 H3); + simplify in H3; + change in H1 with + (∀x1,x2:T1.r x2 x1 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2)); + simplify in H; + apply H; + [ apply H1; + assumption + | assumption + ] + ] + | intros (T1 r Hs m1 m2 H1 t1 t3 H3); + simplify in H3; + change in H1 with + (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2)); + simplify in H; + apply H; + [ apply H1; + assumption + | assumption + ] + | intro v; + elim v 0; + [ intros (T1 r m1 m2 H1 t1 t3 H3); + simplify in H3; + change in H1 with + (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2)); + | intros (T1 r m1 m2 H1 t1 t3 H3); + simplify in H3; + change in H1 with + (∀x1,x2:T1.r x2 x1 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2)); + ] ; + simplify in H; + apply H; + [1,3: + apply H1; + assumption + |2,4: + assumption + ] + | intros (T m1 m2 H1 t1 t3 H3); + simplify in H3; + change in H1 with + (∀x:T. make_compatibility_goal_aux n Out (m1 x) (m2 x)); + rewrite > H3; + simplify in H; + apply H; + [ apply H1 + | assumption + ] + ] + ] +qed. +(* Theorem apply_morphism_compatibility_Left2Right: ∀In Out (m1 m2: function_type_of_morphism_signature In Out) (args1 args2: product_of_arguments In). @@ -755,4 +860,4 @@ Add Morphism not with signature impl -→ impl as Not_Morphism2. unfold impl; tauto. Qed. -*) \ No newline at end of file +*) -- 2.39.2